↑

Properties

Label 2-1379-1379.797-c0-0-0
Degree $2$
Conductor $1379$
Sign $0.183 + 0.983i$
Analytic cond. $0.688210$
Root an. cond. $0.829584$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.66 − 0.868i)2-s + (1.44 − 2.06i)4-s + (−0.462 + 0.886i)7-s + (0.365 − 2.83i)8-s + (−0.284 + 0.958i)9-s + (0.372 − 1.91i)11-s + 1.87i·14-s + (−0.981 − 2.66i)16-s + (0.358 + 1.84i)18-s + (−1.04 − 3.50i)22-s + (−0.180 + 0.406i)23-s + (−0.718 + 0.695i)25-s + (1.16 + 2.23i)28-s + (−0.00205 − 0.0640i)29-s + (−1.83 − 1.66i)32-s + ⋯
L(s)  = 1  + (1.66 − 0.868i)2-s + (1.44 − 2.06i)4-s + (−0.462 + 0.886i)7-s + (0.365 − 2.83i)8-s + (−0.284 + 0.958i)9-s + (0.372 − 1.91i)11-s + 1.87i·14-s + (−0.981 − 2.66i)16-s + (0.358 + 1.84i)18-s + (−1.04 − 3.50i)22-s + (−0.180 + 0.406i)23-s + (−0.718 + 0.695i)25-s + (1.16 + 2.23i)28-s + (−0.00205 − 0.0640i)29-s + (−1.83 − 1.66i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.183 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1379\)    =    \(7 \cdot 197\)
Sign: $0.183 + 0.983i$
Analytic conductor: \(0.688210\)
Root analytic conductor: \(0.829584\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1379} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1379,\ (\ :0),\ 0.183 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.574660900\)
\(L(\frac12)\) \(\approx\) \(2.574660900\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.462 - 0.886i)T \)
197 \( 1 + (-0.991 - 0.127i)T \)
good2 \( 1 + (-1.66 + 0.868i)T + (0.572 - 0.820i)T^{2} \)
3 \( 1 + (0.284 - 0.958i)T^{2} \)
5 \( 1 + (0.718 - 0.695i)T^{2} \)
11 \( 1 + (-0.372 + 1.91i)T + (-0.926 - 0.375i)T^{2} \)
13 \( 1 + (-0.462 - 0.886i)T^{2} \)
17 \( 1 + (0.518 + 0.855i)T^{2} \)
19 \( 1 + (0.900 - 0.433i)T^{2} \)
23 \( 1 + (0.180 - 0.406i)T + (-0.672 - 0.740i)T^{2} \)
29 \( 1 + (0.00205 + 0.0640i)T + (-0.997 + 0.0640i)T^{2} \)
31 \( 1 + (0.871 - 0.490i)T^{2} \)
37 \( 1 + (0.601 - 1.63i)T + (-0.761 - 0.648i)T^{2} \)
41 \( 1 + (-0.518 - 0.855i)T^{2} \)
43 \( 1 + (-1.41 - 0.274i)T + (0.926 + 0.375i)T^{2} \)
47 \( 1 + (-0.801 + 0.598i)T^{2} \)
53 \( 1 + (1.89 - 0.121i)T + (0.991 - 0.127i)T^{2} \)
59 \( 1 + (-0.159 + 0.987i)T^{2} \)
61 \( 1 + (-0.284 - 0.958i)T^{2} \)
67 \( 1 + (-0.627 - 1.88i)T + (-0.801 + 0.598i)T^{2} \)
71 \( 1 + (1.57 + 0.466i)T + (0.838 + 0.545i)T^{2} \)
73 \( 1 + (-0.761 - 0.648i)T^{2} \)
79 \( 1 + (-0.0480 + 0.118i)T + (-0.718 - 0.695i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.871 + 0.490i)T^{2} \)
97 \( 1 + (0.949 - 0.315i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.893868971918528277651808432850, −8.936453503437031227720564583485, −7.995952152554981975230333400694, −6.64129593446691286681819312837, −5.75860997702026311190448182553, −5.55933829482110263530944409210, −4.42232982339624455169488214751, −3.30560587390700116170970676988, −2.84689127853913916052922345131, −1.62630756870947996591836728431, 2.18718509061285301231523679374, 3.45286391991540539726812984942, 4.14382041164216811643832122105, 4.73827882552189644610775442961, 5.93794763586159259227083907525, 6.55918251840819699510607674050, 7.22991779826354493591022717741, 7.77959194401061573450756543438, 9.104670600912539822919448984795, 9.929893534872839409135619145255

Graph of the $Z$-function along the critical line